algorithms and complexity 2: complexity notation

John Levine, Computer and Information Sciences, Strathclyde Universityhttp://www.cis.strath.ac.uk/~johnl/

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Algorithms and Complexity2: Complexity Notation

John [email protected]


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Last time

• What’s an algorithm then?– Intro and brief history– Sample use of algorithms– Sample algorithm - room count

• Course overview– Lectures, practical, labs/tuts, the exam


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The travelling seller• Mary, a computer seller, needs to visit 3 shops in a day (starting and finishing at the office): what’s the shortest route?

• What if there’s 12 shops?

5km

3km

2km

8km

12km


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Today

• Size matters

• Complexity notation

• Calculating complexities

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Download time from internet

• Say it takes 2s to establish a link and then we download at 5K/s, then if n is size of file in K:– time to download is

n/5 + 2

– a linear function

– dominant element as data size grows is n, so using big-oh notation = O(n)


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String Search

• Problem:– Find a string t in a longer string s

• Simple approach i=0; found = false;

while (!found) & (i<s.length)

j = 0; oksofar = true;

while (oksofar) & (j<t.length())

if (s.charAt(i+j)!=t.charAt(j)) oksofar = false;

j++

found = oksofar; i++;


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Rules for big-oh notation

• Only record the highest, dominant, factor

• Ignore constants and multipliers

• Kind of like rounding• 1,000,001 is roughly 1,000,000• so you say O(n2) and not O(12n2)

• 4n3 + 3n2 + 1000n + 2000000 = O(n3)

• 3n3 + 5 = O(n3)


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Common classes of Big-Oh

• In increasing complexity roughly:- logarithmic O(log n)

- linear O(n)

- lin-log O(n log n)

- quadratic O(n2)

- polynomial O(nk), k > 1

- exponential O(an), n > 1

- factorial O(n!), n > 1


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Common classes of Big-Oh

0

100

200

300

400

500

600

700

800

900

1000

0 20 40 60 80 100 120

number

time

log n

n

n log n

n*n

2*n*n

n!


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Tyranny of complexity

• 1000 items per sec plus 10 secs startupn O(log n) O(n) O(n2) O(n3) O(n!) 1 10 sec 10 sec 10 sec 10 sec 10 sec 2 10 sec 10 sec 10 sec 10 sec 10 sec 4 10 sec 10 sec 10 sec 10 sec 10 sec 8 10 sec 10 sec 10 sec 11 sec 50 sec

16 10 sec 10 sec 10 sec 14 sec 663 years 32 10 sec 10 sec 11 sec 43 sec 8.34E+24 years 64 10 sec 10 sec 14 sec 5 min 4.02E+78 years

128 10 sec 10 sec 26 sec 35 min 1.22E+205 years 256 10 sec 10 sec 1 min 5 hours 512 10 sec 11 sec 5 min 37 hours

1,024 10 sec 11 sec 18 min 12 days 2,048 10 sec 12 sec 1 hours 3 months 4,096 10 sec 14 sec 5 hours 2 years 8,192 10 sec 18 sec 19 hours 17 years

16,384 10 sec 26 sec 3 days 139 years 32,768 10 sec 43 sec 12 days 1113 years 65,536 10 sec 1 min 2 months 8901 years

131,072 10 sec 2 min 6 months 71209 years 262,144 10 sec 5 min 2 years 569672 years 524,288 10 sec 9 min 9 years 4557377 years

1,048,576 10 sec 18 min 35 years 36,459,013 yrs


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<= = >=

• Big-Oh says “as the numbers grow the dominant factor will be no worse (<=) than given”– e.g. an O(n log n) function grows no faster

than another t = n log n

• Big-Oh is used for the worst case - we will mostly talk worst, sometimes expected but never best nor average

• Also Ω(n) and θ(n) for >= and =


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Simple String Search

Peter Piper picked a peck of pickled pepper; A peck of pickled pepper Peter Piper picked. If Peter Piper picked a peck of pickled pepper, Where’s the peck of pickled pepper Peter Piper picked?

• Find pepper - lots of false starts

• Can you do better?

• Simple complexity is O(nm)


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Better string search• Start search at end and keep a tablepeter_piper picked a peck of pickled pepper

pepperpiper_picked a peck of pickled pepper

peter pepperpicked a peck of pickled pepper

peter piper pepper a peck of pickled pepper

peter piper pickedpepperk of pickled pepper

peter piper picked a pecpepperickled pepper

peter piper picked a peck pepperkled pepper

peter piper picked a peck of picpepperepper

peter piper picked a peck of picklpepperper

peter piper picked a peck of pickledpepperr

peter piper picked a peck of pickled pepppeppeerr

10 steps instead of 50 (I think)

what about longer text?

what about fewer letters - e.g. DNA coding?

AGCCCGAACATTTACGCCGCTGGCGACTGCACCG


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String search

• Basic algorithm is O(nm)

• Best algorithm is O(n)– BUT is much more complex

• Have to think of when it matters– is data big enough for more complex soln– watch constants and multipliers


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Summary

• Size matters

• Complexity notation

• Calculating complexities

• Next time: start searching & sorting

• Ask the class quiz

algorithms and complexity 2: complexity notation

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